Search for anisotropic birefringent spacetime-symmetry breaking in gravitational wave propagation from GWTC-3

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Search for anisotropic, birefringent spacetime-symmetry breaking in gravitational

wave propagation from GWTC-3

Le¨ıla Haegel∗

Universit´e Paris Cit´e, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France

Kellie O’Neal-Ault†and Quentin G. Bailey‡

Embry-Riddle Aeronautical University, Prescott, AZ, 86301, USA

Jay D. Tasson and Malachy Bloom

Carleton College, Northﬁeld, MN 55057, USA

Lijing Shao

Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China

National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China

(Dated: September 8, 2023)

An eﬀective ﬁeld theory framework, the Standard-Model Extension, is used to investigate the

existence of Lorentz and CPT-violating eﬀects during gravitational wave propagation. We implement

a modiﬁed equation for the dispersion of gravitational waves, that includes isotropic, anisotropic

and birefringent dispersion. Using the LIGO-Virgo-KAGRA algorithm library suite, we perform a

joint Bayesian inference of the source parameters and coeﬃcients for spacetime symmetry breaking.

From a sample of 45 high conﬁdence events selected in the GWTC-3 catalog, we obtain a maximal

bound of 3.19×10−15 m at 90% CI for the isotropic coeﬃcient k(5)

(V)00 when assuming the anisotropic

coeﬃcients to be zero. The combined measurement of all the dispersion parameters yields limits

on the order of 10−13 m for the 16 k(5)

(V)ij coeﬃcients. We study the robustness of our inference by

comparing the constraints obtained with diﬀerent waveform models, and ﬁnd that a lack of physics

in the simulated waveform may appear as spacetime symmetry breaking-induced dispersion for a

subset of events.

I. INTRODUCTION

In the search for a fundamental uniﬁed theory of

physics, it may be imperative to reconsider the axioms

underlying General Relativity (GR) and the Standard

Model (SM) of particle physics. Many theoretical propos-

als argue for a possible breaking of spacetime symmetries,

including Lorentz invariance (LI) and Charge-Parity-

Time (CPT) symmetry [1–6], in such a way that it may

be detectable in sensitive tests. The direct detections

of gravitational waves (GWs) reported by the LIGO-

Virgo-KAGRA (LVK) collaboration provide a new chan-

nel to test the rich phenomenology induced by spacetime-

symmetry breaking in the gravitation sector [7–11].

The eﬀective ﬁeld theory referred to as the Standard-

Model Extension (SME) is a theoretical framework dedi-

cated to derive the observable consequences of spacetime-

symmetry breaking that is punctilious and model in-

dependent. The framework is comprised of the action

of GR and the SM plus all possible terms obtainable

from GR and SM ﬁeld operators contracted with coeﬃ-

cients for spacetime-symmetry breaking, including local

Lorentz, CPT, and diﬀeomorphism breaking terms [12–

21]. Extensive constraints have been derived on these

∗l.haegel@ip2i.in2p3.fr

†aultk@my.erau.edu

‡baileyq@erau.edu

terms within the matter sector and in the gravity sec-

tor [22], the latter having been studied with a wide range

of astrophysical probes [15, 19, 20, 23–25]. Existing anal-

ysis includes short-range gravity tests [26–28], gravimetry

tests [29–35], astrophysical tests with pulsars [36–38], so-

lar system planetary tests [39–41], near-Earth tests [42–

45], and tests with GWs [19, 46–51]. We complement

those searches with further study of the LI and CPT-

violating eﬀects on propagation of GW. We use dynam-

ical equations for the metric ﬂuctuations derived from

the action of the SME, and the resulting eﬀects include

dispersion, anisotropy, and birefringence.

Several tests of GR have been performed with the GW

events detected by the LVK [47–50, 52–54]. Some re-

lated works focus on parameterizations of the deviations

from GR [9, 10, 55, 56], including waveform consistency

tests, modiﬁcation of the GW generation, presence of

extra polarization modes, and tests using speciﬁc mod-

els [55, 57–62]. The current searches for LI violation

performed by the LVK collaboration notably rely on a

modiﬁed dispersion relation that includes isotropic and

polarization-independent eﬀects [53, 63, 64]. Using the

SME framework, we extend this phenomenology by mea-

suring the coeﬃcients for LI and CPT violation, including

anisotropic and polarization-dependent dispersion. First

estimates of those coeﬃcients have been derived using

posterior probabilities released with previous GW catalog

releases, eﬀectively neglecting the correlations between

the parameters describing the source and the spacetime-

arXiv:2210.04481v2 [gr-qc] 7 Sep 2023

symmetry breaking coeﬃcients [49, 50]. In this article,

we present a joint measurement of the source parameters

and the coeﬃcients, alongside studying the robustness of

the results we obtain.

Section II summarizes the derivation of the phe-

nomenology induced by LI and CPT violation in the

SME framework. Section III details the methodologi-

cal aspects, including the dataset used for the measure-

ment of the spacetime-symmetry breaking coeﬃcients.

Section IV presents the obtained results, as well as a

discussion of the impact of the underlying gravitational

waveform model and correlations with source parame-

ters. Section V discusses those results in light of existing

studies and future GW instrument sensitivities. Theo-

retical portions of this paper work with natural units,

where ℏ=c= 1 and Newton’s gravitational constant is

GN̸= 1, while our data analysis work follows SI units.

Greek letters are used for spacetime indices while Latin

letters for spatial indices. We work with the spacetime

metric signature (−,+,+,+).

II. THEORETICAL DERIVATION OF A

DISPERSION RELATION FOR

GRAVITATIONAL WAVES

We summarize previous derivations in [19, 65, 66], fo-

cusing on gravity-sector terms within the SME frame-

work. The spacetime metric is expanded as ﬂuctuations

about the Minkowski metric, gµν =hµν +ηµν , and we

consider up to second order in hµν for the action, which is

suﬃcient to characterize propagation eﬀects. This gives

the following action:

I=1

8κZd4x hµν ˆ

K(d)µνρσ hρσ.(1)

The operator ˆ

K(d)µνρσ , consists of partial derivatives

that act on hµν ,

K(d)µνρσ =K(d)µνρσϵ1...ϵd−2∂ϵ1...∂ϵd−2,(2)

and K(d)µνρσϵ1...ϵd−2are general background coeﬃcients

that are considered small, constant and control the size

of any Lorentz or CPT violation;

Ensuring linearized gauge symmetry, i.e., hµν →hµν +

∂µξν+∂νξµ, and retaining only terms that contribute to

the resulting ﬁeld equations, we arrive at the following

Lagrange density [19]:

L=1

8κϵµρακϵνσβληκλhµν ∂α∂βhρσ

8κhµν (ˆsµρνσ + ˆqµρνσ +ˆ

kµρνσ )hρσ.(3)

The ﬁrst term is the standard GR term written with the

totally antisymmetric Levi-Civita tensor density ϵµρακ,

and the remaining terms contain all additional Lorentz

invariant and violating terms, organized into three terms

based on symmetry properties: ˆsis CPT even with mass

dimension d≥4; ˆqis CPT odd with mass dimension d≥

5; ˆ

kis CPT even with mass dimension d≥6. Details of

these terms including the corresponding Young Tableaux

can be found in Table 1 of Ref. [19]. As an example, for

mass dimension 5,

ˆqµρνσ =q(5)µρϵνζσκ∂ϵ∂ζ∂κ,(4)

where q(5)µρϵνζσκ has 60 independent components. Note

that the gauge symmetry requirement can be relaxed [20,

67], but we do not consider such terms here. The ori-

gin of the eﬀective action for hµν resulted from explicit

symmetry breaking or spontaneous-symmetry breaking

as discussed elsewhere [15, 19, 68, 69].

Performing the variation with respect to hµν on the

action (3), results in the vacuum ﬁeld equations,

0 = Gµν −[1

4(ˆsµρνσ + ˆsµσνρ) + 1

2ˆ

kµνρσ

8(ˆqµρνσ + ˆqνρµσ + ˆqµσνρ + ˆqνσµρ)] hρσ .(5)

Assuming plane wave solutions, ¯

hµν =Aµν e−ipαxα,

where xµis spacetime position and pµ= (ω, ⃗p) is the

four-momentum for the plane wave, and transforming

into momentum space with ∂α=−ipαthe dispersion

relation can be obtained independently of gauge condi-

tions, as shown in references [20, 70]. The dispersion

relation for the two propagating modes is given by

ω=|⃗p|1−ζ0± |⃗

ζ|,(6)

where

|⃗

ζ|=p(ζ1)2+ (ζ2)2+ (ζ3)2(7)

and

ζ0=1

4|⃗p|2−ˆsµν µν +1

2ˆ

kµν µν ,

(ζ1)2+ (ζ2)2=1

8|⃗p|4ˆ

kµνρσ ˆ

kµνρσ −ˆ

kµρ νρ ˆ

kµσ νσ

8ˆ

kµν µν ˆ

kρσ ρσ,

(ζ3)2=1

16|⃗p|4−1

2ˆqµρνσ ˆqµρνσ −ˆqµνρσ ˆqµνρσ

+(ˆqµρν ρ+ ˆqνρµ ρ)ˆqµσν σ).(8)

We retrieve the GR case when symmetry-breaking co-

eﬃcients, i.e. ζ0and |⃗

ζ|, vanish. Note that this result

holds at leading order in the coeﬃcients for Lorentz vi-

olation, hence higher modes do not contribute in this

perturbative treatment [20, 70]. Relaxing some of the

assumptions in this framework, allowing for other ﬁelds

to contribute dynamically to the action, could result in

additional modes [71, 72].

GR predicts two linearly independent polarizations for

GWs propagating in vacuum, traveling at the speed of

light. Possible modiﬁcations for observable Lorentz and

CPT violating eﬀects from (6) include birefringence, e.g.,

altered relative travel speeds between the polarizations,

which result from the two possible signs for |⃗

ζ|in (6), re-

quiring a minimum mass dimension 5. Furthermore, the

presence of higher powers of frequency and momentum

in the terms above, indicates beyond GR dispersion as

well. All of these eﬀects depend on the sky location of

the propagating wave, and thus a breaking of rotational

isotropy occurs.

To take into account the sky localization dependence

of the source in the detector frame, it is advantageous

to project the SME coeﬃcients onto spherical harmon-

ics [70],

ζ0=X

djm

ωd−4Yjm(ˆ

n)k(d)

(I)jm,(9)

ζ1∓i ζ2=X

djm

ωd−4±4Yjm(ˆ

n)k(d)

(E)jm ±ik(d)

(B)jm,(10)

ζ3=X

djm

ωd−4Yjm(ˆ

n)k(d)

(V)jm,(11)

where −j≤m≤j, the Yjm(ˆ

n) are the standard spheri-

cal harmonics while ±4Yjm(ˆ

n) are spin-weighted spheri-

cal harmonics, and ˆn=−ˆp.

Expressions for the two linearly independent GW po-

larizations, in the transverse-traceless gauge, result in a

phase shift from the additional symmetry-breaking ef-

fects,

h(+) =eiδ(cos β−isin ϑcos φsin β)hLI

(+)

−eiδ sin β(cos ϑ+isin ϑsin φ)hLI

(×)

h(×)=eiδ(cos β+isin ϑcos φsin β)hLI

(×)

+eiδ sin β(cos ϑ−isin ϑsin φ)hLI

(+).(12)

where

δ=ωd−3τζ(d)0,

β=ωd−3τ|⃗

ζ(d)|,

and the modiﬁed redshift becomes

τ=Zz

dz (1 + z)d−4

H(z).(13)

For notational convenience, the angles in (12) are deﬁned

by the expressions below,

sin ϑ=|ζ1∓iζ2|

|⃗

ζ|,cos ϑ=ζ3

|⃗

ζ|, e∓iφ =ζ1∓iζ2

√(ζ1)2+(ζ2)2.

(14)

One of the key features of spacetime symmetry break-

ing, as evidenced in the equations above, is the breaking

of isotropy. The strength of the LI violation can change

with source location [66]. Unless otherwise stated, the

spherical coeﬃcients in (9)-(11) are expressed in the Sun-

Centered Celestial Equatorial reference frame (SCF), as

is standard in the literature [22, 73], and allows com-

parisons with other non-GW tests in the gravity sector.

Rotations and boosts of the spherical coeﬃcients relative

to this frame must be taken into account, as discussed

elsewhere [74].

III. DATA AND PARAMETER INFERENCE

A. Bayesian inference of source and

symmetry-breaking parameters

For mass dimension 4, LI violation leads to a modiﬁ-

cation of the GW group velocity that can be measured

with multimessenger signals; constraints on the ˆsopera-

tor of Eq. (3) have been obtained from the observation of

GW170817/GRB170817A [47] comparing light and GW

travel time, and travel time across the Earth [48].

In this analysis, we focus on the coeﬃcients for Lorentz

and CPT violation contained in the operator ˆqfor d= 5

(see Eq (5)), with the ﬁrst mass dimension in the action

series (3) where GW dispersion occurs. We probe the

impact of isotropic and anisotropic dispersion, as well

as birefringence, with a joint estimation of the source

parameters and the 16 a priori independent k(5)

(V)ij coef-

ﬁcients of Eq. (11). Speciﬁcally, we are considering in

this work, a subset of (12), where δ= 0. In this case the

remaining coeﬃcients are contained in β. The expression

is lengthy but takes the form [66]:

β(5) =ω2τ(5)

2√π

k(5)

(V)00 −q3

2sin θeiϕ k(5)

(V)11 +e−iϕ k(5)∗

(V)11

+√3 cos θ k(5)

(V)10 +...

,(15)

with the superscript (5) meaning all quantities are eval-

uated with d= 5 like equation (13).

Using a Bayesian inference framework, we compare the

strain detected by the LVK interferometers with a tem-

plate bank of gravitational waveforms modiﬁed as out-

lined in Eq. (12). The strain takes the form,

SA=F(+)h(+) +F(×)h(×),(16)

where h(+,×)are the expressions (12), and F(+,×)are the

standard detector response functions. The rotation an-

gles relating diﬀerent frames, are included in the expres-

sions for F(+,×). These are deﬁned in the LALSuite soft-

ware, including the source frame and the detector frame.

Again, the coeﬃcients k(5)

(V)ij in (15) are left in the SCF.

We use the LALSuite algorithm package, modifying the

LALSimulation subpackage to generate dispersed wave-

forms and performing the parameter estimation with a

custom version of LALInference [75]. For a single event,

LALInference evaluates the posterior probability with a

Markov chain process using the matched-ﬁltered likeli-

hood:

P(d|⃗

θGR,⃗

θSM E , I) = exp X

i−2|˜

di−˜

hi(⃗

θGR ,⃗

θSM E )|2

T Sn(fi)

−1

2log πT Sn(fi)

2

(17)

where ˜

hiis the template signal, ˜

diis the interferome-

ter datastream, Tis the duration of the signal, and Sn

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Searchforanisotropic,birefringentspacetime-symmetrybreakingingravitationalwavepropagationfromGWTC-3Le¨ılaHaegel∗Universit´eParisCit´e,CNRS,AstroparticuleetCosmologie,F-75013Paris,FranceKellieO’Neal-Ault†andQuentinG.Bailey‡Embry-RiddleAeronauticalUniversity,Prescott,AZ,86301,USAJayD.TassonandMalachyB...

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Search for anisotropic birefringent spacetime-symmetry breaking in gravitational wave propagation from GWTC-3

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